Hi man! A fractal might do the trick up my sleeve!

Welcome to rustcircle. I am a mathematician and programmer, and this is where I write about mathematics, artificial intelligence and quantum computing — with the occasional chess detour.

The picture behind the text#

The soft pattern moving behind this page is a Julia set. Fix a complex number $c$ and iterate

z_{n+1} = z_n^2 + c, \qquad z_0 = z \in \mathbb{C}.

The filled Julia set $K_c$ is the set of starting points whose orbit stays bounded. Here $c$ travels slowly around the circle $|c| = 0.7885$ , so the set morphs continuously:

c(t) = 0.7885\, e^{i\theta(t)}, \qquad \theta(t) = \theta_0 + \omega t .

Colour comes from the smooth escape time $\nu(z) = n - \log_2 \log_2 |z_n|$ , which removes the banding of the integer iteration count. Everything runs in one tiny WebGL shader at half resolution and 30 frames per second, and it stops entirely if you ask your system for reduced motion.

Fractal plasma, one of the oldest tricks in 80s computer graphics — Diamond–square *plasma fractal*, the texture of many 80s demos.

A small piece of code#

def escape_time(z: complex, c: complex, n_max: int = 56) -> int:
    for n in range(n_max):
        if abs(z) > 16:
            return n
        z = z * z + c
    return n_max

What comes next#

The Endless Quest! Or infinite adventures!
Perhaps you already know or else we both might imagine!
A world full of wonders!